Implementation Aspects in Regularized Structural Equation Models

Abstract

This article reviews several implementation aspects in estimating regularized single-group and multiple-group structural equation models (SEM). It is demonstrated that approximate estimation approaches that rely on a differentiable approximation of non-differentiable penalty functions perform similarly to the coordinate descent optimization approach of regularized SEMs. Furthermore, using a fixed regularization parameter can sometimes be superior to an optimal regularization parameter selected by the Bayesian information criterion when it comes to the estimation of structural parameters. Moreover, the widespread penalty functions of regularized SEM implemented in several R packages were compared with the estimation based on a recently proposed penalty function in the Mplus software. Finally, we also investigate the performance of a clever replacement of the optimization function in regularized SEM with a smoothed differentiable approximation of the Bayesian information criterion proposed by O’Neill and Burke in 2023. The findings were derived through two simulation studies and are intended to guide the practical implementation of regularized SEM in future software pieces.

Keywords

• structural equation modeling
• confirmatory factor analysis
• regularized estimation
• Bayesian information criterion